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A352064 Irregular triangle T(n,k) where row n lists the positions of n in A275314.

%I A352064 #14 Mar 04 2022 11:20:16
%S A352064 1,2,3,4,6,8,5,9,12,16,10,18,24,32,7,15,20,27,36,48,64,14,30,40,54,72,
%T A352064 96,128,21,25,28,45,60,80,81,108,144,192,256,42,50,56,90,120,160,162,
%U A352064 216,288,384,512,11,35,63,75,84,100,112,135,180,240,243,320,324,432,576,768,1024
%N A352064 Irregular triangle T(n,k) where row n lists the positions of n in A275314.
%C A352064 A table by Leonhard Euler.
%C A352064 Let L(n-1) be a partition of (n-1) whose parts m are restricted to predecessors of primes. Row n lists the products (m+1) for all such partitions L(n-1).
%C A352064 Greatest term in row n is 2^(n-1).
%C A352064 Least term in row p prime is p.
%H A352064 Michael De Vlieger, <a href="/A352064/b352064.txt">Table of n, a(n) for n = 1..10673</a> (rows n = 1..40, flattened)
%H A352064 Leonhard Euler, <a href="https://archive.org/stream/tentamennovaethe00eule#page/40/mode/2up">Tentamen novae theoriae mvsicae ex certissimis harmoniae principiis dilvcide expositae</a>, Petropoli, ex typographia Academiae scientiarvm (1739), 41.
%F A352064 A280954(n) = length of row n.
%e A352064 Triangle begins:
%e A352064    1;
%e A352064    2;
%e A352064    3,  4;
%e A352064    6,  8;
%e A352064    5,  9, 12, 16;
%e A352064   10, 18, 24, 32;
%e A352064    7, 15, 20, 27,  36,  48,  64;
%e A352064   14, 30, 40, 54,  72,  96, 128;
%e A352064   21, 25, 28, 45,  60,  80,  81, 108, 144, 192, 256;
%e A352064   42, 50, 56, 90, 120, 160, 162, 216, 288, 384, 512;
%e A352064   ...
%e A352064 Illustration of relationship of terms of row n and partitions of (n-1) such that all parts m are restricted to prime predecessors. We show the partitions in parentheses, adding 1 to each part m below in brackets to take the product. The products are terms in row n in this sequence.
%e A352064       1 = (1);
%e A352064           [2]
%e A352064 row 2:     2;
%e A352064 .
%e A352064       2 = (2),    (1+1);
%e A352064           [3]     [2*2]
%e A352064 row 3:     3,       4;
%e A352064 .
%e A352064       3 = (2+1),  (1+1+1);
%e A352064           [3*2]   [2*2*2]
%e A352064 row 4:     6,       8;
%e A352064 .
%e A352064       4 = (4),    (2+2),    (2+1+1),     (1+1+1+1);
%e A352064           [5]     [3*3]     [3*2*2]      [2*2*2*2]
%e A352064 row 5:     5,       9,        12,           16;
%e A352064 .
%e A352064       5 = (4+1),  (2+2+1),  (2+1+1+1),   (1+1+1+1+1);
%e A352064           [5*2]   [3*3*2]   [3*2*2*2]    [2*2*2*2*2]
%e A352064 row 6:    10,      18,        24,           32;
%e A352064 etc.
%t A352064 With[{n = 12}, Take[#, n] &@ Values@ KeySort@ PositionIndex@ Array[Total[Flatten[ConstantArray[#1 - 1, #2] & @@@ FactorInteger[#]]] &, 2^n]] // TableForm (* syntactically simple, or, more efficiently *)
%t A352064 f[m_] := Block[{s = {Prime@ PrimePi[m + 1] - 1}}, KeySort@ Merge[#, Identity] &@ Join[{1 -> {}}, Reap[Do[If[# <= m, Sow[# -> s]; AppendTo[s, Last@ s], If[Last@ s == 1, s = DeleteCases[s, 1]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]], s = MapAt[Prime[PrimePi[# + 1] - 1] - 1 &, s, -1]]] &@ Total[s], {i, Infinity}]][[-1, -1]] ]]; Map[Union[Times @@ # & /@ #] &, Values@ f[40] + 1] // Flatten
%Y A352064 Cf. A001414, A006093, A275314, A280954.
%K A352064 nonn,tabf
%O A352064 1,2
%A A352064 _Michael De Vlieger_, Mar 02 2022